Question: Is ${279312}$ divisible by $3$ ?
Explanation: A number is divisible by $3$ if the sum of its digits is divisible by $3$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {279312}= &&{2}\cdot100000+ \\&&{7}\cdot10000+ \\&&{9}\cdot1000+ \\&&{3}\cdot100+ \\&&{1}\cdot10+ \\&&{2}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {279312}= &&{2}(99999+1)+ \\&&{7}(9999+1)+ \\&&{9}(999+1)+ \\&&{3}(99+1)+ \\&&{1}(9+1)+ \\&&{2} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {279312}= &&\gray{2\cdot99999}+ \\&&\gray{7\cdot9999}+ \\&&\gray{9\cdot999}+ \\&&\gray{3\cdot99}+ \\&&\gray{1\cdot9}+ \\&& {2}+{7}+{9}+{3}+{1}+{2} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $3$ , so the first five terms must all be multiples of $3$ That means that to figure out whether the original number is divisible by $3 $ , all we need to do is add up the digits and see if the sum is divisible by $3$ . In other words, ${279312}$ is divisible by $3$ if ${ 2}+{7}+{9}+{3}+{1}+{2}$ is divisible by $3$ Add the digits of ${279312}$ $ {2}+{7}+{9}+{3}+{1}+{2} = {24} $ If ${24}$ is divisible by $3$ , then ${279312}$ must also be divisible by $3$ ${24}$ is divisible by $3$, therefore ${279312}$ must also be divisible by $3$.